A supercharacter theory for the Sylow p-subgroups of the finite symplectic and orthogonal groups

Details A supercharacter theory for the Sylow p-subgroups of the finite symplectic and orthogonal groups A supercharacter theory for the Sylow p-subgroups of the finite symplectic and orthogonal groups

2008-10-21

arxiv.org/abs/0810.3761v2

Mathematics

Group Theory

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Scientific paper

We define the superclasses for a classical finite unipotent group $U$ of type $B_{n}(q)$, $C_{n}(q)$, or $D_{n}(q)$, and show that, together with the supercharacters defined in a previous paper, they form a supercharacter theory. In particular, we prove that the supercharacters take a constant value on each superclass, and evaluate this value. As a consequence, we obtain a factorization of any superclass as a product of elementary superclasses. In addition, we also define the space of superclass functions, and prove that it is spanned by the supercharacters. As as consequence, we (re)obtain the decomposition of the regular character as an orthogonal linear combination of supercharacters. Finally, we define the supercharacter table of $U$, and prove various orthogonality relations for supercharacters (similar to the well-known orthogonality relations for irreducible characters).

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